Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Find the interval in which the function are strictly increasing or strictly decreasing :

$f (x) = x^{4} - 4 x$ .

Accepted Answer

The given function is,

$f (x) = x^{4} - 4 x$

On differentiating both sides w.r.t. $x$ , we get

$\frac{d}{d x} f (x) = \frac{d}{d x} (x^{4} - 4 x)$

$f' (x) = \frac{d}{d x} (x^{4}) - \frac{d}{d x} (4 x)$ $[∵ \frac{d}{d x} [f (x) + g (x)] = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)]$

$f' (x) = 4 x^{3} - 4$ $[∵ \frac{d}{d x} x^{n} = n x^{n - 1}]$

$f' (x) = 4 (x^{3} - 1)$ $[$ Taking common $4]$

$f' (x) = 4 (x - 1) (x^{2} + x + 1)$

$f' (x) = 4 (x - 1) (x^{2} + x + \frac{1}{4} + \frac{3}{4})$

$f' (x) = 4 (x - 1) ({(x + \frac{1}{2})}^{2} + \frac{3}{4})$

For $f (x)$ is increasing, we get

$f' (x) > 0$

$\Rightarrow 4 (x - 1) ({(x + \frac{1}{2})}^{2} + \frac{3}{4}) > 0$

$\Rightarrow (x - 1) > 0$ $[∵ {(x + \frac{1}{2})}^{2} + \frac{3}{4} > 0]$

$\Rightarrow x > 1$

$\Rightarrow x \in (1, \infty)$

Hence , $f (x)$ is increasing on $[1, \infty)$

For $f (x)$ is decreasing, we get

$f' (x) < 0$

$\Rightarrow 4 (x - 1) ({(x + \frac{1}{2})}^{2} + \frac{3}{4}) < 0$

$\Rightarrow (x - 1) < 0$ $[∵ {(x + \frac{1}{2})}^{2} + \frac{3}{4} > 0]$

$\Rightarrow x < 1$

$\Rightarrow x \in (- \infty, 1)$

Thus, $f (x)$ is decreasing on $(- \infty, 1]$ .

Find the interval in which the function are strictly increasing or strictly decreasing :

$f (x) = x^{4} - 4 x$ .

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the interval in which the function are strictly increasing or strictly decreasing : fx=x4-4x.

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the interval in which the function are strictly increasing or strictly decreasing :

$f (x) = x^{4} - 4 x$ .